MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isphg Structured version   Visualization version   GIF version

Theorem isphg 30836
Description: The predicate "is a complex inner product space." An inner product space is a normed vector space whose norm satisfies the parallelogram law. The vector (group) addition operation is 𝐺, the scalar product is 𝑆, and the norm is 𝑁. An inner product space is also called a pre-Hilbert space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
Hypothesis
Ref Expression
isphg.1 𝑋 = ran 𝐺
Assertion
Ref Expression
isphg ((𝐺𝐴𝑆𝐵𝑁𝐶) → (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ CPreHilOLD ↔ (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec ∧ ∀𝑥𝑋𝑦𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁𝑥)↑2) + ((𝑁𝑦)↑2))))))
Distinct variable groups:   𝑥,𝑦,𝐺   𝑥,𝑁,𝑦   𝑥,𝑆,𝑦   𝑥,𝑋,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)

Proof of Theorem isphg
Dummy variables 𝑔 𝑛 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ph 30832 . . 3 CPreHilOLD = (NrmCVec ∩ {⟨⟨𝑔, 𝑠⟩, 𝑛⟩ ∣ ∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔(((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛𝑥)↑2) + ((𝑛𝑦)↑2)))})
21elin2 4203 . 2 (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ CPreHilOLD ↔ (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec ∧ ⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ {⟨⟨𝑔, 𝑠⟩, 𝑛⟩ ∣ ∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔(((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛𝑥)↑2) + ((𝑛𝑦)↑2)))}))
3 rneq 5947 . . . . . 6 (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺)
4 isphg.1 . . . . . 6 𝑋 = ran 𝐺
53, 4eqtr4di 2795 . . . . 5 (𝑔 = 𝐺 → ran 𝑔 = 𝑋)
6 oveq 7437 . . . . . . . . . 10 (𝑔 = 𝐺 → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
76fveq2d 6910 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑛‘(𝑥𝑔𝑦)) = (𝑛‘(𝑥𝐺𝑦)))
87oveq1d 7446 . . . . . . . 8 (𝑔 = 𝐺 → ((𝑛‘(𝑥𝑔𝑦))↑2) = ((𝑛‘(𝑥𝐺𝑦))↑2))
9 oveq 7437 . . . . . . . . . 10 (𝑔 = 𝐺 → (𝑥𝑔(-1𝑠𝑦)) = (𝑥𝐺(-1𝑠𝑦)))
109fveq2d 6910 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑛‘(𝑥𝑔(-1𝑠𝑦))) = (𝑛‘(𝑥𝐺(-1𝑠𝑦))))
1110oveq1d 7446 . . . . . . . 8 (𝑔 = 𝐺 → ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2) = ((𝑛‘(𝑥𝐺(-1𝑠𝑦)))↑2))
128, 11oveq12d 7449 . . . . . . 7 (𝑔 = 𝐺 → (((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑠𝑦)))↑2)))
1312eqeq1d 2739 . . . . . 6 (𝑔 = 𝐺 → ((((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛𝑥)↑2) + ((𝑛𝑦)↑2))) ↔ (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑠𝑦)))↑2)) = (2 · (((𝑛𝑥)↑2) + ((𝑛𝑦)↑2)))))
145, 13raleqbidv 3346 . . . . 5 (𝑔 = 𝐺 → (∀𝑦 ∈ ran 𝑔(((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛𝑥)↑2) + ((𝑛𝑦)↑2))) ↔ ∀𝑦𝑋 (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑠𝑦)))↑2)) = (2 · (((𝑛𝑥)↑2) + ((𝑛𝑦)↑2)))))
155, 14raleqbidv 3346 . . . 4 (𝑔 = 𝐺 → (∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔(((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛𝑥)↑2) + ((𝑛𝑦)↑2))) ↔ ∀𝑥𝑋𝑦𝑋 (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑠𝑦)))↑2)) = (2 · (((𝑛𝑥)↑2) + ((𝑛𝑦)↑2)))))
16 oveq 7437 . . . . . . . . . 10 (𝑠 = 𝑆 → (-1𝑠𝑦) = (-1𝑆𝑦))
1716oveq2d 7447 . . . . . . . . 9 (𝑠 = 𝑆 → (𝑥𝐺(-1𝑠𝑦)) = (𝑥𝐺(-1𝑆𝑦)))
1817fveq2d 6910 . . . . . . . 8 (𝑠 = 𝑆 → (𝑛‘(𝑥𝐺(-1𝑠𝑦))) = (𝑛‘(𝑥𝐺(-1𝑆𝑦))))
1918oveq1d 7446 . . . . . . 7 (𝑠 = 𝑆 → ((𝑛‘(𝑥𝐺(-1𝑠𝑦)))↑2) = ((𝑛‘(𝑥𝐺(-1𝑆𝑦)))↑2))
2019oveq2d 7447 . . . . . 6 (𝑠 = 𝑆 → (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑠𝑦)))↑2)) = (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑆𝑦)))↑2)))
2120eqeq1d 2739 . . . . 5 (𝑠 = 𝑆 → ((((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑠𝑦)))↑2)) = (2 · (((𝑛𝑥)↑2) + ((𝑛𝑦)↑2))) ↔ (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑛𝑥)↑2) + ((𝑛𝑦)↑2)))))
22212ralbidv 3221 . . . 4 (𝑠 = 𝑆 → (∀𝑥𝑋𝑦𝑋 (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑠𝑦)))↑2)) = (2 · (((𝑛𝑥)↑2) + ((𝑛𝑦)↑2))) ↔ ∀𝑥𝑋𝑦𝑋 (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑛𝑥)↑2) + ((𝑛𝑦)↑2)))))
23 fveq1 6905 . . . . . . . 8 (𝑛 = 𝑁 → (𝑛‘(𝑥𝐺𝑦)) = (𝑁‘(𝑥𝐺𝑦)))
2423oveq1d 7446 . . . . . . 7 (𝑛 = 𝑁 → ((𝑛‘(𝑥𝐺𝑦))↑2) = ((𝑁‘(𝑥𝐺𝑦))↑2))
25 fveq1 6905 . . . . . . . 8 (𝑛 = 𝑁 → (𝑛‘(𝑥𝐺(-1𝑆𝑦))) = (𝑁‘(𝑥𝐺(-1𝑆𝑦))))
2625oveq1d 7446 . . . . . . 7 (𝑛 = 𝑁 → ((𝑛‘(𝑥𝐺(-1𝑆𝑦)))↑2) = ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2))
2724, 26oveq12d 7449 . . . . . 6 (𝑛 = 𝑁 → (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)))
28 fveq1 6905 . . . . . . . . 9 (𝑛 = 𝑁 → (𝑛𝑥) = (𝑁𝑥))
2928oveq1d 7446 . . . . . . . 8 (𝑛 = 𝑁 → ((𝑛𝑥)↑2) = ((𝑁𝑥)↑2))
30 fveq1 6905 . . . . . . . . 9 (𝑛 = 𝑁 → (𝑛𝑦) = (𝑁𝑦))
3130oveq1d 7446 . . . . . . . 8 (𝑛 = 𝑁 → ((𝑛𝑦)↑2) = ((𝑁𝑦)↑2))
3229, 31oveq12d 7449 . . . . . . 7 (𝑛 = 𝑁 → (((𝑛𝑥)↑2) + ((𝑛𝑦)↑2)) = (((𝑁𝑥)↑2) + ((𝑁𝑦)↑2)))
3332oveq2d 7447 . . . . . 6 (𝑛 = 𝑁 → (2 · (((𝑛𝑥)↑2) + ((𝑛𝑦)↑2))) = (2 · (((𝑁𝑥)↑2) + ((𝑁𝑦)↑2))))
3427, 33eqeq12d 2753 . . . . 5 (𝑛 = 𝑁 → ((((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑛𝑥)↑2) + ((𝑛𝑦)↑2))) ↔ (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁𝑥)↑2) + ((𝑁𝑦)↑2)))))
35342ralbidv 3221 . . . 4 (𝑛 = 𝑁 → (∀𝑥𝑋𝑦𝑋 (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑛𝑥)↑2) + ((𝑛𝑦)↑2))) ↔ ∀𝑥𝑋𝑦𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁𝑥)↑2) + ((𝑁𝑦)↑2)))))
3615, 22, 35eloprabg 7543 . . 3 ((𝐺𝐴𝑆𝐵𝑁𝐶) → (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ {⟨⟨𝑔, 𝑠⟩, 𝑛⟩ ∣ ∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔(((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛𝑥)↑2) + ((𝑛𝑦)↑2)))} ↔ ∀𝑥𝑋𝑦𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁𝑥)↑2) + ((𝑁𝑦)↑2)))))
3736anbi2d 630 . 2 ((𝐺𝐴𝑆𝐵𝑁𝐶) → ((⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec ∧ ⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ {⟨⟨𝑔, 𝑠⟩, 𝑛⟩ ∣ ∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔(((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛𝑥)↑2) + ((𝑛𝑦)↑2)))}) ↔ (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec ∧ ∀𝑥𝑋𝑦𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁𝑥)↑2) + ((𝑁𝑦)↑2))))))
382, 37bitrid 283 1 ((𝐺𝐴𝑆𝐵𝑁𝐶) → (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ CPreHilOLD ↔ (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec ∧ ∀𝑥𝑋𝑦𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁𝑥)↑2) + ((𝑁𝑦)↑2))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  cop 4632  ran crn 5686  cfv 6561  (class class class)co 7431  {coprab 7432  1c1 11156   + caddc 11158   · cmul 11160  -cneg 11493  2c2 12321  cexp 14102  NrmCVeccnv 30603  CPreHilOLDccphlo 30831
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-cnv 5693  df-dm 5695  df-rn 5696  df-iota 6514  df-fv 6569  df-ov 7434  df-oprab 7435  df-ph 30832
This theorem is referenced by:  cncph  30838  isph  30841  phpar  30843  hhph  31197
  Copyright terms: Public domain W3C validator